Well, an option for an Orthodox Golden Horde is to have the Romans be impressive - it is one of the reasons Islam (AFAIK) was adopted. Now, the Romans had converted people to Orthodoxy before, notably, the Russians.
I suggest an initial PoD (with a few others) where Manuel has success in the Morea and conquers it - avoiding the Battle of Prinitza's humiliation. With that campaign a success, Manuel can afford to hire more Seljuk Turks into his forces, giving him a stronger cavalry arm for later expansion - which prepares him for Nogai in 1265 with a stronger army. If this can lead to an unexpected victory for the Romans (perhaps they bring the Bulgars onto their side, or some other strategem), and the capture of Nogai, then he can be 'released' as a 'guest' and brought to Constantinople, and given the Hagia Sofia treatment (and hopefully conversion). A peace treaty later where Constantinople pays Nogai tribute, and marries off in exchange for the Romans getting control over Bulgaria and a treaty that puts south of the Danube under Roman, not Mongol rule. If Nogai can work his way out of that situation, perhaps by being able to commit more forces because of Manuel - perhaps the OTL marriage still takes place, which could have .. interesting consequences.
If Nogai is able to turn his defeat into the above described diplomatic victory, he could certainly rebuild his reputation and rather than commit his efforts to conquering south of the Danube, instead focuses on taking control of the Golden Horde - which with Roman help (and a few cities in recompense here and there), you could have the Golden Horde be Orthodox via Nogai, with strong Roman influence, later supplanted by a Mongol priesthood.
The right circumstances, and a well timed crisis, and you could see the Nogai as Roman Emperors, which is always a fun twist.